Can a Derivative Trade Have a Positive Credit Valuation Adjustment but a Negative Funding Valuation Adjustment? ▴ Question

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Concept

The question of whether a single derivative position can simultaneously possess a positive Credit Valuation Adjustment (CVA) and a negative Funding Valuation Adjustment (FVA) cuts to the core of modern derivatives pricing. It moves beyond the textbook Black-Scholes framework into the operational realities of a post-2008 financial system. The answer is an unequivocal yes, and understanding this duality is fundamental to grasping how risk and cost are priced into uncollateralized financial instruments. This scenario is not an anomaly; it is the standard state for any uncollateralized derivative that represents an asset to the calculating entity, such as a bank.

At its heart, the situation reveals two distinct and separate risks that a bank must manage. CVA is the market price of the counterparty’s potential default. When a bank holds a derivative that has a positive value, it is effectively a receivable; the counterparty owes the bank money.

The CVA quantifies the risk that the counterparty will fail to pay, representing a necessary charge to compensate for bearing that specific credit risk. A positive CVA is a debit to the trade’s overall value from the bank’s perspective, a risk mitigation charge passed on to the counterparty.

CVA prices the counterparty’s credit risk, while FVA prices the bank’s own funding costs for the position.

Conversely, FVA addresses a completely different operational challenge ▴ the cost of funding. When the bank hedges its position in the market, those hedges are typically executed under standard, fully collateralized terms. If the uncollateralized derivative with the client is an asset, the bank’s corresponding hedges are likely liabilities, requiring the bank to post collateral. This creates a cash outflow.

The bank must borrow funds to cover this collateral posting, and the cost of that borrowing, determined by the bank’s own creditworthiness in the funding markets, is the FVA. Because this is a cost ▴ a cash outflow to the bank’s treasury ▴ it is recorded as a negative adjustment to the derivative’s value.

Therefore, the coexistence of a positive CVA and a negative FVA is the logical financial outcome for a derivative asset. The bank charges for the client’s risk (positive CVA) while simultaneously accounting for its own internal cost to carry the position (negative FVA). This dynamic reflects the systemic reality that every financial instrument carries at least two layers of risk ▴ the risk of who you are trading with and the operational cost of how you must fund your side of the trade.

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Strategy

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The Duality of Counterparty Risk and Funding Cost

Strategically dissecting the CVA and FVA dynamic requires viewing a derivative not as a single price, but as a bundle of risks and costs that must be independently valued and managed. The core scenario ▴ a positive CVA alongside a negative FVA ▴ arises from the fundamental asymmetry in an uncollateralized trade between a financial institution and a counterparty, such as a corporate client. This asymmetry is rooted in access to funding and the nature of market-standard hedging practices.

Consider a bank entering into a five-year, uncollateralized interest rate swap with a corporate client. The bank agrees to pay a fixed rate and receive a floating rate. If interest rates rise, the value of the swap becomes positive for the bank; it is now an asset on the bank’s books. This immediately triggers the CVA/FVA duality.

CVA Generation ▴ The positive value means the bank has a credit exposure to the corporate client. The client is obligated to make future payments that are now more valuable. The bank’s CVA desk must quantify the risk of the client defaulting on these obligations over the remaining life of the trade. This calculation results in a CVA charge, which is economically a price for the credit guarantee the bank is extending. This charge increases the price quoted to the client and represents a positive adjustment in the bank’s valuation framework.
FVA Generation ▴ The bank does not leave its market risk unhedged. It enters the interbank market to execute an opposing swap, paying floating and receiving fixed. This hedge is executed under a standard Credit Support Annex (CSA), meaning it is fully collateralized. As the bank’s client swap is an asset, the hedge swap is a liability. The bank must now post collateral to its interbank counterparty. To acquire the cash for this collateral, the bank’s treasury must borrow at its own funding rate. This borrowing incurs a cost over the life of the trade. The FVA is the present value of this expected future funding cost, a direct negative adjustment to the profitability of the client trade.

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Systemic Drivers of CVA and FVA Divergence

The magnitude of the CVA and FVA is driven by different sets of market variables. A trading desk’s strategy must involve monitoring and managing these drivers, as they directly impact the profitability of the derivatives portfolio. The divergence between a positive CVA and a negative FVA is a function of these independent economic forces.

The table below outlines the primary drivers for each valuation adjustment from the perspective of a bank holding a derivative asset.

Valuation Adjustment	Primary Driver	Impact on Adjustment	Strategic Implication
Credit Valuation Adjustment (CVA)	Counterparty Credit Spread	A wider spread (higher default risk) increases the positive CVA.	The bank must charge more for taking on the counterparty’s credit risk. Pricing becomes less competitive for lower-quality credits.
Funding Valuation Adjustment (FVA)	Bank’s Own Funding Spread	A wider spread (higher borrowing cost for the bank) increases the negative FVA.	The bank’s own credit health directly impacts its profitability. A less stable bank finds it more expensive to run its derivatives book.
Credit Valuation Adjustment (CVA)	Market Volatility	Higher volatility increases the potential future exposure (PFE), thus increasing the positive CVA.	Trades on more volatile underlyings will carry higher CVA charges, reflecting the greater uncertainty in future exposure.
Funding Valuation Adjustment (FVA)	Expected Future Exposure (EFE)	A higher positive EFE requires more funding for hedges, increasing the negative FVA.	The structure of the derivative itself (e.g. its potential to become a large asset) dictates the scale of the funding cost.

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The Collateralization Mitigation Strategy

The most effective strategy for managing and mitigating this CVA/FVA divergence is collateralization. The introduction of a two-way CSA, where both parties agree to post collateral against the mark-to-market value of the derivative, fundamentally alters the risk landscape. A fully collateralized trade, where collateral is posted daily, would theoretically have a CVA and FVA of zero. This is because collateral eliminates the credit exposure (mitigating CVA) and the funding asymmetry (mitigating FVA), as the cash received from the client trade can be used to fund the hedge trade.

Collateralization is the primary mechanism to neutralize both counterparty credit risk and its associated funding costs.

However, many corporate clients do not have the operational capacity or desire to enter into fully collateralized agreements. This creates a market for uncollateralized trades, where the bank’s ability to accurately price and manage CVA and FVA becomes a competitive advantage. The strategic decision for a bank is not whether to eliminate these adjustments, but how to price them correctly and hedge them efficiently across a portfolio of trades.

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Execution

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Quantitative Mechanics of XVA Calculation

The execution of a derivatives trade valuation requires a precise quantitative framework for calculating each component of the XVA spectrum. While the full models are complex, involving Monte Carlo simulations over many risk factors, their core logic can be understood through simplified formulas. These formulas illustrate the mechanics that lead to a positive CVA and a negative FVA for a derivative asset held by a bank.

Credit Valuation Adjustment (CVA) ▴ The CVA is the sum of discounted expected losses over the life of the trade. It is driven by the potential future exposure to the counterparty. A simplified representation is ▴ CVA = (1 - R) Σ
- (1 – R) ▴ This is the Loss Given Default (LGD), representing the portion of the exposure that will not be recovered in bankruptcy. R is the recovery rate.
- EPE(t) ▴ The Expected Positive Exposure at a future time t. This is the average of all simulated positive values of the derivative at that time, representing the bank’s potential claim.
- PD(t, t+Δt) ▴ The marginal probability of the counterparty defaulting in the time interval between t and t+Δt, derived from their credit default swap (CDS) spreads.
- DF(t) ▴ The discount factor to bring the future expected loss back to its present value.
Funding Valuation Adjustment (FVA) ▴ The FVA is the sum of discounted funding costs or benefits. For a derivative asset, it is a cost. A simplified representation is ▴ FVA = Σ
- EPE(t) ▴ The same Expected Positive Exposure. The amount of exposure dictates how much hedging is required and, therefore, how much cash must be borrowed to post as collateral.
- Spread_Bank ▴ The bank’s own funding spread over the risk-free rate, representing its cost of borrowing.
- Spread_Collateral ▴ The rate earned on posted cash collateral, often a risk-free overnight rate. The net funding cost is the difference.
- Δt ▴ The length of the time interval.
- DF(t) ▴ The discount factor.

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Illustrative Trade Scenario a Numerical Walkthrough

To make this tangible, consider a hypothetical $100 million, 5-year uncollateralized interest rate swap between a Bank (AA-rated) and a Corporate (BBB-rated). The swap is an asset for the Bank. The table below provides a simplified, year-by-year calculation demonstrating the positive CVA and negative FVA.

Year (t)	Expected Positive Exposure (EPE) ($M)	Counterparty PD (%)	CVA Contribution ($)	Bank Funding Spread (%)	FVA Contribution ($)
1	1.50	1.00	15,000	0.50	-7,500
2	2.50	1.20	30,000	0.55	-13,750
3	2.80	1.50	42,000	0.60	-16,800
4	2.00	1.80	36,000	0.65	-13,000
5	1.00	2.00	20,000	0.70	-7,000
Total (Undiscounted)	–	–	+143,000	–	-58,050

The numerical example confirms the trade generates a total positive CVA charge of $143,000 while simultaneously incurring a negative FVA cost of $58,050.

This table (which omits discounting and LGD for clarity) clearly shows the dynamic. Each year, the potential exposure to the BBB-rated corporate generates a CVA charge. Concurrently, the need to fund the hedges for this exposure at the bank’s own funding spread generates an FVA cost.

The total valuation adjustment for the bank would be the sum of these two figures, along with other potential adjustments like DVA (Debit Valuation Adjustment). The execution of pricing for the client must incorporate both of these components to be economically viable for the bank.

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References

Brigo, Damiano, Massimo Morini, and Andrea Pallavicini. Counterparty Credit Risk, Collateral and Funding ▴ With Pricing Cases for All Asset Classes. John Wiley & Sons, 2013.
Gregory, Jon. The xVA Challenge ▴ Counterparty Credit Risk, Funding, Collateral, and Capital. 4th ed. John Wiley & Sons, 2020.
Hull, John, and Alan White. “LIBOR vs. OIS ▴ The Derivatives Discounting Dilemma.” Journal of Investment Management, vol. 11, no. 3, 2013, pp. 14-27.
Castagna, Antonio. “The xVA Challenge ▴ A P&L and Balance Sheet Perspective.” Applied Mathematical Finance, vol. 25, no. 1, 2018, pp. 1-27.
Burgard, Christoph, and Mats Kjaer. “In the Balance.” Risk Magazine, vol. 24, no. 7, 2011, pp. 72-75.
Pallavicini, Andrea, Daniele Perini, and Damiano Brigo. “Funding, Collateral, and Hedging ▴ Uncovering the Mechanics and the Subtleties of Modern Credit and Funding Pricing.” SSRN Electronic Journal, 2012.
Kenyon, Chris, and Andrew Green. “XVA ▴ Pricing, Risk, and Hedging.” In Handbook of Financial Engineering, edited by Constantine Georgiou, et al. World Scientific, 2021, pp. 201-245.
Crépey, Stéphane. “Bilateral Counterparty Risk under Funding Constraints ▴ Part I ▴ A Comprehensive Modeling Framework.” Mathematical Finance, vol. 25, no. 1, 2015, pp. 1-22.

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Reflection

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A System of Interconnected Valuations

The recognition that a single instrument houses both a positive CVA and a negative FVA moves valuation from a static calculation to a dynamic systems analysis. It forces an institution to look inward at its own funding architecture and outward at the credit profile of its counterparties with equal scrutiny. The final price of a derivative is not a monolithic number but the result of an intricate interplay between external risks and internal costs. This understanding transforms the pricing process from a simple market-matching exercise into a profound assessment of the firm’s own position within the financial ecosystem.

How does your own operational framework account for this duality? Is the cost of funding treated as a direct, traceable input to pricing, or is it socialized across the enterprise? The answer reveals the maturity of a firm’s risk and valuation systems.

A truly robust framework treats FVA with the same analytical rigor as CVA, recognizing that a mispriced cost of funding is just as damaging as a mispriced credit risk. The ultimate advantage lies not in eliminating these adjustments, but in building a system that can see, price, and manage them with precision.